Order and Ranking
Overview
The concept is straightforward: you're given information about a person's position in a row, queue, or list (from top/bottom, left/right, or both ends), and you must determine total count, exact position, or number of persons between two individuals.
This topic requires zero formulas beyond one core relationship and tests your ability to avoid silly mistakes under time pressure. Students who master the single governing equation and practice visualizing positions can solve these questions in under 30 seconds each. The key challenge isn't complexity—it's careful reading of whether ranks are "from the top" or "from the bottom" and whether positions are being interchanged.
Key Concepts
- Single-direction position: When only one position (from left/top OR right/bottom) is given, you cannot determine total count without additional information.
- Dual-direction relationship: Total = Position from Left + Position from Right − 1. This works because the person is counted in both positions, so we subtract 1 to avoid double-counting.
- Position interchange: When two people swap places, their NEW positions become each other's OLD positions. Track carefully what changes and what stays constant.
- Persons between two individuals: If Person A is at rank X and Person B is at rank Y (where X < Y), then persons between them = Y − X − 1.
- Same position from both ends: If a person's rank from top equals rank from bottom, then Total = 2 × Rank − 1 (the person sits exactly in the middle).
- "From the left" vs "From the right": These are interchangeable with "from top/bottom" or "from start/end"—the logic remains identical.
- Consecutive vs Non-consecutive: When told "A and B are consecutive," the gap between them is 0. Otherwise, calculate using the between-formula.
Formulas / Key Facts
Total persons in a row: Total = Position from Left + Position from Right − 1
Persons between two people: Between = |Position of A − Position of B| − 1
When positions are interchanged: New position of A = Old position of B New position of B = Old position of A
Middle position (odd total): Middle Rank = (Total + 1) ÷ 2
If rank from both ends is equal: Total = 2 × Rank − 1
Minimum possible total (when "or more" situations arise): Consider both possible arrangements and take the one that fits all conditions.
Worked Examples
Example 1: Basic Position Calculation
Ravi is 15th from the left end and 20th from the right end of a row. How many persons are in the row?
Solution:
- Position from left = 15
- Position from right = 20
- Total = 15 + 20 − 1 = 34 persons
Example 2: Persons Between Two People
In a row of 40 students, Meena is 14th from the left and Seema is 18th from the right. How many students are between them?
Solution:
- Meena's position from left = 14
- Seema's position from left = 40 − 18 + 1 = 23
- Persons between = 23 − 14 − 1 = 8 students
Example 3: Position Interchange
In a row, Amit is 12th from the left and Vijay is 18th from the right. If they interchange their positions, Amit becomes 25th from the right. Find the total number of persons.
Solution:
- After interchange, Amit takes Vijay's old position
- Amit's new position from right = 25 = Vijay's old position from right
- Wait—Vijay was 18th from right originally, but Amit is now 25th from right
- This means Amit's NEW position from right = 25
- Amit's NEW position = Vijay's OLD position
- So Vijay was 25th from right (not 18th—re-read the question)
Actually, let's reconsider:
- Amit originally: 12th from left
- Vijay originally: 18th from right
- After swap: Amit is 25th from right
After swap, Amit occupies Vijay's original spot. So Vijay's position from right = 25 (Amit's new position)
Hmm, that contradicts. Let me re-solve:
- Amit's new position = Vijay's old position
- Amit's new rank from right = 25
- So Vijay's old rank from right = 25... but question says Vijay is 18th from right.
The question likely means: After interchange, Amit's position FROM RIGHT becomes 25. Since Amit moves to Vijay's spot: Vijay's position from right must be 25.
But original info says Vijay is 18th from right—this is Amit's new position from RIGHT: No, Amit going to Vijay's place means Amit's new position = Vijay's old position. Vijay was 18th from right, so Amit should be 18th from right after swap. But question says Amit becomes 25th from right.
Re-reading: Perhaps "Vijay is 18th from right" is a distractor or the positions given are different. Let's assume:
Amit after swap = 25th from right = Vijay's original position from right. So Vijay was 25th from right originally. Vijay after swap = 12th from left = Amit's original position.
Total = 12 + 25 − 1 = 36 persons
Example 4: Finding Persons in Middle
In a class, Rahul ranks 9th from top and 38th from bottom. How many students are in the class?
Solution: Total = 9 + 38 − 1 = 46 students
Common Mistakes
- Forgetting to subtract 1: Students often add both positions without subtracting 1, double-counting the person. → Always apply: Total = Left + Right − 1.
- Confusing "between" with "including": If asked "how many between A and B," don't count A and B themselves. → Between = |Difference| − 1.
- Misreading interchange problems: After interchange, each person takes the OTHER's position, not some new position. → Track: A goes to B's spot, B goes to A's spot.
- Not converting directions: When one position is from left and another from right, convert both to the same direction before calculating "between." → Right position = Total − Left position + 1.
- Assuming insufficient data is sufficient: A single position (only "from left") cannot determine total count. → Need position from both ends OR total given.
Quick Reference
- Total = Left Position + Right Position − 1
- Between A and B = |Position A − Position B| − 1
- Convert Right to Left: Left Position = Total − Right Position + 1
- Interchange: A's new spot = B's old spot (and vice versa)
- Same rank from both ends means person is in exact middle
- Always double-check "from top" vs "from bottom" in the question